The boundedness of the weighted Coxeter group with complete graph (Q2821716)

Let \(W\) be a (not necessarily finite) Coxeter group and let \(S\) be its set of Coxeter generators. Lusztig has previously defined a weight function on the Coxeter system \((W,S)\) that we denote by \(L\) and call the triple \((W,S,L)\) weighted Coxeter group. Writing \(\{c_w\mid w\in W\}\) for Lusztig's basis for the corresponding Iwahori-Hecke algebra, the coefficients \(h_{x,y,z}\in\mathbb{Z}[v,v^{-1}]\) are defined by the expression \(c_xc_y=\sum_{z\in W}h_{x,y,z}c_z\). We then say that \((W,S,L)\) is bounded if the degrees of the \(h_{x,y,z}\)s are bounded. In this paper, the authors provide the following sufficient condition for being bounded. Given \(s,t\in S\) we write \(w_{st}\) for the longest element of the subgroup generated by \(s\) and \(t\), \(\langle s,t\rangle\). A weighted Coxeter group is bounded if the corresponding Coxeter graph is complete and the set NEWLINE\[NEWLINE \{L(u),L(w_{st})\mid u,s,t\in W\text{ and }\langle s,t\rangle\} NEWLINE\]NEWLINE is bounded.